The condition \(\textrm {Re}_{\rm mag}^{\rm ini} \gg 1\)  in combination with Eqs. (REF ) and () bounds the hypermagnetic energy density at the end of inflation from below.
A second, upper bound follows from the chiral plasma instability [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, which is based on the chiral magnetic effect [8]}, [9]}, [10]}, [11]}.
For our purposes, the essence of this effect is that large charge asymmetries in the chiral SM fermion species will trigger an instability in gauge-field modes that carry opposite helicity compared to the total helicity of the background field.
This will wash out the helicity stored in the hypermagnetic field as well as the charge asymmetries stored in the chiral fermions.
